Question

The diameter of an electric cable, Say X, is assumed to be continuous random variable with p.d.f . f(x) =6x (1-x), 0 less than equal to x less than equal to 1.
Show that () fx is a p.d.f.
B. Determine a number b such that P(xb)
C. Find the mean of X.

Answer 1

Given :  The diameter of an electric cable, Say X, is assumed to be continuous random variable with p.d.f f (x)= 6x(1-x),    0 ≤ x  ≤ 1

To find : Show that f(x)  is a p.d.f.  (probability density function )

Solution:

f(x)  = 6x(1 - x)    0 ≤ x  ≤ 1

Function  f(x)  is a probability density function  in range a  to b

if     $\int\limits^b_a {f(x)} \, dx = 1$

f(x)  = 6x(1 - x)  

f(x) = 6x  - 6x²

$\int\limits^1_0 {(6x - 6x^2)} \, dx$

$= [ \frac{6x^2}{2} - \frac{6x^3}{3} ] ^1_0$

$= [ 3x^2 - 2x^3 ] ^1_0$

= 3(1)² - 2(1)³  - ( 0 - 0)

= 3 - 2 - 0

= 1

$\int\limits^1_0 {(6x - 6x^2)} \, dx = 1$

Hence f(x) = 6x(1 - x)    0 ≤ x  ≤ 1  is pdf  ( probability density function )

Mean =  $\int\limits^1_0 x{(6x - 6x^2)} \, dx$

$= [ \frac{6x^3}{3} - \frac{6x^4}{4} ] ^1_0$

$= [{2x^3} - \frac{3x^4}{2} ] ^1_0$

= 2(1)³ - 3(1)⁴/2  - ( 0 -0 )

= 2  - 3/2 - 0

= 1/2

Mean of X = 1/2

Learn more:

A random variable X has the probability distribution - Brainly.in

brainly.in/question/5372170

p.d.f . f(x) =6x (1-x)

https://brainly.in/question/17508018

Answer 2

Answer:

I hope you understand the solution

Step-by-step explanation: